JCE does not create the read more link / button, it only allows you to place a marker in the content where the read more should be created, which is done by the Joomla Article Manager when the article is rendered in the front-end.

To change the final position of this button, you would need to create a template override of the blog_item page - you can read about template overrides here - https://www.joomlashack.com/blog/joomla-templates/customize-your-joomla-templates-by-learning-overrides/ - and a layout override of the readmore itself.